I’m going to talk today about a little bit of algebra. There’s a beautiful relation to algebraic topology which I’ll touch on as well. (For a brief introduction to cup products, refer to section 7 of my notes here: http://www.mit.edu/~sanathd/papers/algebraic-topology.pdf.)
Definition: A Hopf algebra is a graded -algebra satisfying:
- There is an identity such that given by is an isomorphism.
- There is a homomorphism of graded algebras , called the diagonal/coproduct, such that for and , we have: , where .
For example: . Because elements of of dimension lower than are the elements of , the terms and are , and the diagonal is hence .
Another example is the algebra where is a field. In this case, is primitive, i.e., . Therefore, we have:
But because we’re thinking of the truncated polynomial ring, , and hence:
Thus if , we must have . If is of characteristic zero this is impossible. If has nonzero characteristic , then must be a power of (this can be proved using the fact that the freshman’s dream is true in characteristic ).
Yet another example is the exterior algebra over with an odd-dimensional generator. (This is the free -module with basis generated by with multiplication defined by . For example, the cohomology ring .) This is a Hopf algebra with diagonal ; to show that it indeed forms a Hopf algebra we must only check that . But , so consider . This is equal to . The latter term is since has odd dimension, and by definition, so we’re done.
One more rather important example: if is a finitely generated free -module in each dimension then is a Hopf algebra, called the dual Hopf algebra (see Proposition 3C.10 in Hatcher). The product is the dual of the diagonal , and the diagonal is the dual of the product .
Ok, so now onto topology. It turns out that there are natural examples of Hopf algebras in algebraic topology. We need two pieces of background. First, the notion of a H-space. Recall the notion of a topological group; this is a space with a group structure such that the product and the inverse map are continuous. Every Lie group is a topological group, for example. This motivates:
Definition: A H-space is a space with a continuous multiplication and identity such that and are homotopic to the identity.
For example, every topological group is a H-space. Another classic and very important example is a loop space , whose product is:
Note that this isn’t necessarily commutative. Now, suppose is a H-space such that:
- , which is implied, for example, if is path-connected.
- is finitely generated for each , so the cross product is an isomorphism.
The multiplication induces , giving a map:
This is a map of -algebras. Let be the inclusion , and consider the commutative diagram:
The map satisfies and for . By commutativity, , and hence . Hence the component of in the tensor is . Continuing like so, we get the following formula for where are of degree :
And we get the structure of a Hopf algebra! This is very cool. Let’s use the examples of Hopf algebras we computed early on to look at examples of H-spaces. So for finite , we see that is not a H-space by the example above about truncated polynomial rings. But, because polynomial rings are Hopf algebras, is a H-space! Recall from homology:
Proposition: The boundary map in is given by . The product of a boundary and cycle is a boundary because and if . This gives a bilinear map , and hence a homomorphism , which is the cross product in homology.
If is a H-space, the composition is called the Pontryagin product. This isn’t associative unless the product is associative up to homotopy. We then have a natural example of dual Hopf algebras coming from algebraic topology (the result follows from the fact that the hypotheses imply that .)!
Theorem: If is a H-space whose multiplication is associative up to homotopy and is a finitely generated free -module for all , then with the Pontryagin product is the dual Hopf algebra of .
What about ; is it a H-space? If we choose in the example about truncated polynomial rings, we see that we must have a power of . The universal cover of a path-connected H-space is a H-space, and Adams showed that are the only spheres which are H-spaces. Since for gets their H-space structures from , and thus can be a H-space only for . A pretty cool result, huh? This depends on Adams’ theorem (the Hopf invariant one theorem) on the values of for which has a H-space structure, which was proved using K-theory! I plan on writing a post about this in the near future.